group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the Tits group ²''F''₄(2)′, named for

Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and ...

(), is a finite

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

of order : 2¹¹ · 3³ · 5² · 13 = 17,971,200. It is sometimes considered a 27th

sporadic group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

History and properties

The

Ree groups In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a dif ...

²''F''₄(2^2''n''+1) were constructed by , who showed that they are simple if ''n'' ≥ 1. The first member of this series ²''F''₄(2) is not simple. It was studied by who showed that it is almost simple, its

derived subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

²''F''₄(2)′ of index 2 being a new simple group, now called the Tits group. The group ²''F''₄(2) is a

group of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...

and has a

BN pair BN, Bn or bn may refer to: Businesses and organizations * RTV BN, a Bosnian Serb TV network * Bangladesh Navy * Barisan Nasional (also known as "National Front"), a political coalition in Malaysia * Barnes & Noble, an American specialty retaile ...

, but the Tits group itself does not have a

. Because the Tits group is not strictly a group of Lie type, it is sometimes regarded as a 27th

.For instance, by the

ATLAS of Finite Groups The ''ATLAS of Finite Groups'', often simply known as the ''ATLAS'', is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational assistance from J. ...

and it
web-based descendant
/ref> The

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...

of the Tits group is trivial and its

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...

has order 2, with the full automorphism group being the group ²''F''₄(2). The Tits group occurs as a maximal subgroup of the Fischer group Fi₂₂. The groups ²''F''₄(2) also occurs as a maximal subgroup of the

Rudvalis group In the area of modern algebra known as group theory, the Rudvalis group ''Ru'' is a sporadic simple group of order : 214335371329 : = 145926144000 : ≈ 1. History ''Ru'' is one of the 26 sporadic groups and was found by and c ...

, as the point stabilizer of the rank-3 permutation action on 4060 = 1 + 1755 + 2304 points. The Tits group is one of the simple N-groups, and was overlooked in

John G. Thompson John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008. ...

's first announcement of the classification of simple ''N''-groups, as it had not been discovered at the time. It is also one of the

thin finite group In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number ''p'', the Sylow ''p''-subgroups of the 2- local subgroups are cyclic. Informally, these are the groups that resem ...

s. The Tits group was characterized in various ways by and .

Maximal subgroups

and independently found the 8 classes of maximal subgroups of the Tits group as follows: L₃(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations. 2. ⁸5.4 Centralizer of an involution. L₂(25) 2². ⁸S₃ A₆.2² (Two classes, fused by an outer automorphism) 5²:4A₄

Presentation

The Tits group can be defined in terms of generators and relations by :

4 = ((ab)^4 ab^)^6 = 1, \,

where 'a'', ''b''is the

commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...

''a''⁻¹''b''⁻¹''ab''. It has an

outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...

obtained by sending (''a'', ''b'') to (''a'', ''b''(''ba'')⁵''b''(''ba'')⁵).

Notes

References

* * * * * * *{{Citation , last1=Wilson , first1=Robert A. , author-link1=Robert A. Wilson (mathematician) , title=The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits , doi=10.1112/plms/s3-48.3.533 , mr=735227 , year=1984 , journal=

Proceedings of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

, series=Third Series , issn=0024-6115 , volume=48 , issue=3 , pages=533–563

External links

ATLAS of Group Representations — The Tits Group
Sporadic groups de:Gruppe vom Lie-Typ#Die Tits-Gruppe